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COUNTING OMITTED VALUES
GRAHAM LORD
Temple University, Philadelphia, Peon. 10122
INTRODUCTION
H. L. Alder in [l] has extended J. L. Brown, J r . Ts result on complete sequences [2]
by showing that if { P j } .
1 9
is a non-decreasing sequence and { k i } . _
is a s e -
quence of positive integers, then with PA = 1 every natural number can be represented as
E
•
ar. P . ,
l
i
i=l
where 0 £ a. < k. if and only if
P .k+1
... <
1 +
y\. p.
i=l
for n = 1, 2, ' " .
He also proves for a given sequence {k^}._ 1
non-decreasing sequence of positive integers \ P j / - _ - |
9
9
there is only one
for which the representation is
unique for every natural number s namely the set { p . } = {^i} •_-,
9
. . . where $ t = 1, 0 2
=
1 + lq, 03 = ( l + k j M l + ka), • • • , 0. = (l + k ^ d + kg) ••• ( l + k . ^ ) , • • • .
This paper investigates those natural numbers not represented by the form
i=l
for 0 < a. ^ k. where {k.} is as above and { P 1 } . _ 1
1
1
A
1
1—X, A 5 "
sequence of positive integers satisfying
n
(1)
P
n + 1^
1+
ZkiPI
i=l
443
n = 1, 2,
is a necessarily increasing
444
COUNTING OMITTED VALUES
When specialized to k. = 1 for all
i
[Nov.
the results obtained include those in Hoggatt T s and
1
Peterson s paper [ 3 ] .
2.
Theorem 1.
UNIQUENESS OF REPRESENTATION
For P . satisfying (1), the representation of the natural number
N
as
2>ipi •
i=l
where 0 < a. < k. is unique.
Proof.
Let N be the smallest integer with possibly two representations
N B
EaBPB
=
S=l
l>tPt '
t=l
where a
f 0 f jS
n
m
If m / n assume m > 11. Then by (1)
SaP
s
s
< Y
^ k P < Pn+1
^ - 1 < Pm - 1 < V
Vp. - 1< V
VP, .
/ J s s
/ ^ t t
jL-d t t
s=l
s=l
Thus, m = n.
t=l
Either a
n
t=l
> |3 or a < B . Suppose without loss of generality a ^. B .
&
J
n
n
n
^
n
*n
The natural number
n-1
E Pf *
p
t t
n-1
=
t=l
a
y^
x ^
s
p
s
+
&n - r pn ')
n
s=l
and since it is l e s s than N it has only one representation.
• • • , n, i . e . ,
p
Hence a
= j3
for
s = 1, 2,
N has a unique representation.
3.
Definition.
OMITTED VALUES
For x > 0 let M(x) be the number of natural numbers l e s s than or equal
to x which a r e not represented by
1973]
COUNTING OMITTED VALUES
445
2>ipi
i=l
Theorem 2. If
n
N
/
JI
i i
n '
*
i=l
is the largest representable integer not exceeding the positive number x then
n
M(x) = [x] - ^ V i
i=l
where [ ]
Proof.
'
is the greatest integer function.
By Theorem 1, it is sufficient to show the number,
integers not exceeding x,
R(x), of representable
equals
XX^i
i=l
But R(x) = R(N) from the definition of N. Now all integers of the form
X>i p i
i=l
with the only restriction that 0 < j3 < a a r e l e s s than N since:
J
n
n
n
n-1
E
i=l
/3.P. < y ^ k . P. + |3 P
i=l
< (1 l + j8 J} p
-1
n
n
n-1
< a. P
n
+>
n
JL-JI
i=l
OL.V.
i i
= N
446
COUNTING OMITTED VALUES
[Nov.
Again by the uniqueness of representation to form
E^pi-
0 < B < a
*n
n
i=l
there are a
•••,
choices for j3 , (l l + k AJ choices for j3 ^, L( l + k n\J
n
.n
n-l
.n-1
n-2
and {l + k j choices for jS^ ; in all there a r e # <>
/
numbers.
choices for j3 _,
*n-2
It remains to count numbers of the form
i
n
P
n
n-1
+ \ " * p. P .
/ ^ *i
i
i=l
which do not exceed
n-1
N = a P + > a. P.
n n x ^ i i
n n
i=l
That is the number of integers
n-1
n-1
i=l
i=l
Hence
R
(SaiPi)
=
Vn
+ R
(Zl a i P i
and because R I ^ P J } = at = af1^)1 then
< nn
\
n
\i=l
/
i=l
[The representable positive integers l e s s than or equal to a 1 P 1 a r e
Pl9
2 P l s *•*,
^P^]
This completes the proof.
As P. is representable, the theorem give M(P.) = P. - $.. and the following result is
immediate.
1973]
COUNTING OMITTED VALUES
447
Corollary.
(
' n
\
n
n
Eaipi)=S"iM(pi)
k i=l
/
=
i=l
ZM("ipi) •
i=l
8
Note that if k. = 1 for all i = 1, 2, ' " , Theorems 3 and 4 in [3] a r e special c a s e s
of the above theorem and corollary.
4.
SOME APPLICATIONS
The two sequences P = F 0 and P = F 0 .,, n = 1, 2, •••
n
2n
n
2n-l
isfy Theorems 1 and 2 for k. = 1 i = 1, 2, • • •. However,
1 + 2(Fo + F y j + * - - + F
v 2
4
2n'
) = F
r
2n+2
+ F
r
2n-l
mentioned in Lf3lJ sat-
- 1 > F
x
~ r 2n+2
with equality only when n = 1, and
1 + 2(FH + • • • + F
) = F
v x
2n-l;
2n+l
+ F
2n-2
+ 1 > F
2n+l
Consequently, by Alder's result,
Theorem 3. Every natural number can be expressed as
2 > i F2i
i=l
and as
Z^iF2i-l •
i=l
where a. and j3. are 0 , 1 , or 2.
To return to the general case, let { k . } be a fixed sequence of positive integers; then
any sequence { P ^ satisfying (1) also satisfies
P
^0
for all n.
PA > 1 = 0i and induction:
P
n-1
n-1
> 1 + ;• k. P. > 1 + T ^ k. (h. = (h
i=l
i=l
This MlOWS from
448
COUNTING OMITTED VALUES
Nov. 1973
Hence by the corollary
n
X"ipi)=Z"i^i-^} *
Ml'
\i=l
/
i=l
with equality iff P, =</).. F u r t h e r m o r e , since k. ^ - 1 then {</>.} is an increasing sequence
and so for every natural number N there exist a, such that
< ^ a i ^ i
i=l
N
Therefore
M(N) < Miy^a. 0. 1 = 0 ,
,i=l
i. e. , N has a representation in the form
2>*i
i=l
These facts, together with Theorem 1, give
Theorem 4. If {k.} is any sequence of positive integers, then every natural
number
has a unique representation as
oo
i=l
where 0 < a. < k. and fa = 1, fa = (1 + k t ) , • • • , </>. = (1 + kj) • • • (1 + k.__1) .
Corollary.
If r is a fixed integer l a r g e r than
unique representation in base
1 then every natural number has a
r.
Proof. In Theorem 4, take 1 + k. = r for all i.
i
5. REFERENCES
1. H. L. Alder,
"The Number System in More General S c a l e s , " Mathematics Magazine,
Vol. 35 (1962), pp. 145-151.
2. John L. Brown, J r . , "Note on Complete Sequences of I n t e g e r s , " The American Math.
Monthly, Vol. 67 (1960), pp. 557-560.
3.
V. E. Hoggatt, J r . , and Brian Peterson, "Some General Results on Representations,"
Fibonacci Quarterly, Vol. 10 (1972), pp. 81-88.